A061382 Decimal expansion of Pi/e.
1, 1, 5, 5, 7, 2, 7, 3, 4, 9, 7, 9, 0, 9, 2, 1, 7, 1, 7, 9, 1, 0, 0, 9, 3, 1, 8, 3, 3, 1, 2, 6, 9, 6, 2, 9, 9, 1, 2, 0, 8, 5, 1, 0, 2, 3, 1, 6, 4, 4, 1, 5, 8, 2, 0, 4, 9, 9, 7, 0, 6, 5, 3, 5, 3, 2, 7, 2, 8, 8, 6, 3, 1, 8, 4, 0, 9, 1, 6, 9, 3, 9, 4, 4, 0, 1, 8, 8, 4, 3, 4, 2, 3, 5, 6, 7, 3, 5, 5, 8, 8, 0, 4, 4, 8
Offset: 1
Examples
Pi/e ~= 1.15572734979092171791009318331269629912...
References
- Paul J. Nahin, "An Imaginary Tale, The Story of [Sqrt(-1)]," Princeton University Press, Princeton, NJ 1998, p. 188.
Links
- Harry J. Smith, Table of n, a(n) for n = 1..20000
- David J. Jeffrey, Problems, Puzzles and Challenges, Maple Transactions Vol. 4 No. 3 (2024).
- Antonio Gracia Llorente, Another Infinite Product for π/e, OSF Preprint, 2024.
- Z. A. Melzak, Infinite Products for Pi*e and Pi/e, The American Mathematical Monthly, Vol. 68, No. 1 (Jan., 1961), pp. 39-41 (3 pages).
- Solution of Problem 11771, The American Mathematical Monthly, 121 (2014).
- Zander, Integrate Integral_{x=0..1} x^(-x) (1-x)^(x-1) sin(Pi x) dx, MathStackExchange Apr. 2013.
Programs
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Mathematica
RealDigits[ Pi/E, 10, 110][[1]] (* Robert G. Wilson v, Jan 15 2004 *)
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PARI
{ default(realprecision, 20080); x=Pi/exp(1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b061382.txt", n, " ", d)) } \\ Harry J. Smith, Jul 22 2009
Formula
Equals Integral_{-oo..oo} cos(x)/(1 + x^2) dx. - Robert G. Wilson v, Jan 15 2004
Equals Integral_{-oo..oo} cos(x)/(1 + x^2)^2 dx. - Amiram Eldar, Jul 21 2020
From Robert FERREOL, Apr 02 2022: (Start)
Equals Integral_{-oo..oo} x*sin(x)/(1 + x^2) dx.
Equals 2*Integral_{-oo..oo} x*sin(x)/(1 + x^2)^2 dx. (End)
Equals Product_{k>=1} ((2*k + 1)/(2*k - 1))^(2*k)*(k/(k + 1))^(2*k + 1). - Antonio Graciá Llorente, Mar 30 2024
Equals 1/A061360. - Alois P. Heinz, Jul 14 2024
Equals Integral_{x=0..1} sin(Pi*x)/(x^x*(1 - x)^(1 - x)) dx. (See Zander link). - Peter Luschny, Oct 31 2024